ABSTRACT
This chapter focuses on functionals which are derived from a positive measure space. It shows that a by-product of the extension theory for Daniell spaces is an extension theory for positive measure spaces. The theory of integration on measurable subsets follows satisfactorily from the general theory by means of the operation of ‘restriction’. The chapter presents a theorem and describes integration with respect to such restriction. It describes the relationship of the Riemann integral to the Lebesgue integral. The disadvantages of the Riemann integral are overcome with the introduction of the Lebesgue integral. Since the Lebesgue integral is the Daniell extension of the Riemann integral, it exhibits all the favourable properties enjoyed by any Daniell integral. The chapter also presents Egoroff’s theorem and the concept of convergence in measure as examples for measurable functions.
